Reminders not showing on iPad

Unfortunately you are seeing expected behavior. There was a dialog box when you first opened the reminders app on your phone stating that this was going to happen. I saw it, read it and decided not to upgrade reminders yet.

Here is Apple’s support article about it.

Get ready for the new Reminders app - Apple Support

The fix is to upgrade all of your other devices. This is the way Apple intended it to work. If you update your iPad to the new iPadOS which is the iPad version of iOS 13, reminders will sync for you again.

At version 13, Apple has split iOS into two systems. The one for iPhones continues to be called iOS; version 13 is already out, and version 13.1 comes out on September 24. (That's tomorrow as I write this.)

The new system name is iPadOS, which runs only on iPads (of reasonably recent vintage). It also comes out on September 24 (moved up from September 30). The public beta versions have been called 13 and then 13.1; probably the first released version will be called 13.1.